Spring 2022 Topology Reading Group

TRG is an informal grad student seminar providing a forum for grad students to share what they have been learning about with other students.

In spring 2022 Braeden Reinoso and I co-organised TRG.

February 3rd: Fraser Binns

Links with Knot Floer homology of low rank

I will discuss a lower bound on the rank of knot Floer homology coming from the BRAID invariant. I will then discuss a resulting classification of links with knot Floer homology of low rank. This talk is based on joint work with Subhankar Dey.

February 10th: Braeden Reinoso

Surface diffeomorphisms, braids, and fibered knots, part 1

This is the first in a running series of expository talks which I'll give throughout the semester each time I speak in TRG. I'll try to keep each talk relatively self-contained and introductory, and related to the topics in the title. For the first talk, I'll describe the Nielsen-Thurston classification of surface diffeomorphisms and discuss how it relates to fibered knots. From this perspective, we'll work towards proving that the figure-eight knot is hyperbolic and understanding its fibered monodromy (spoiler: it's the cat map).

February 17th: Braeden Reinoso

Surface diffeomorphisms, braids, and fibered knots, part 2

This time I'll talk about mapping class groups and the Birman-Hilden correspondence. The talk will be casual and expository in nature, and somewhat introductory, like last time. There will be two goals for this talk: to see some simple knots as lifts of simple braids, and to understand some presentations of mapping class groups of surfaces with small genus.

February 24th: Fraser Binns

Heegaard Floer homology via Immersed Curves

I will talk (very informally) about the immersed curves interpretation of Heegaard Floer homology due to Hanselman-J.Rasmussen-Watson. I will discuss the form this invariant takes and discuss how it can be used to give very quick computations, in favourable situations.

March 4th: Qingfeng Lyu

Delta-hyperbolicity and hyperbolic groups

I hope to give a brief introduction to delta hyperbolic spaces, including the basic geometric facts that describe the spaces. Then I might introduce some first applications to hyperbolic groups (e.g. Tit’s alternative), possibly indicating how geometric group theory basically works in this case.

March 17th: Laura Seaberg

Don't talk to me or my SO(n) ever again

(Just kidding, please do.) If you're like me, the first time you heard that groups like SO(n) were manifolds, you didn't think too hard about it. Well, we can do better than that. We will analyze them using Morse-Bott theory, which extends Morse theory to consider particular critical submanifolds. It turns out that a very natural function on SO(n) is Morse-Bott (for all n) and thus uncovers some aspects of the spaces' structures. My goal is to make this talk chill and just a small peek at the weirdness of groups of matrices.

March 24th: Ethan Farber

I’m learning about the Thurston norm!

You read that right. I’ve been (slowly) reading W. Thurston’s paper “A Norm on the Homology of 3-manifolds.” I’ll tell you a bit about what I’ve read so far, including how it fits into math landscape, and probably ask your help filling in some gaps that I don’t yet understand.

March 31st: Gage Martin

Branched double covers, Dehn surgery, Heegaard Floer homology, and Khovanov homology

In 2005 Ozsváth and Szabó constructed a spectral sequence from Khovanov homology to Heegaard Floer homology using a connection between Dehn surgery and branched double covers. In this talk we will aim to illuminate this topological construction and demystify some of the algebraic techniques underlying this and other spectral sequences. No prior background will be assumed.

April 7th: Ali Naseri Sadr

Hilbert-Smith Conjecture

Hilbert conjectured in his fifth problem that every topological group which is also a topological manifold is a Lie group. Yamabe solved this conjecture and Smith generalized it to the following:

If a topological group acts continuously and faithfully on a manifold, the group is a Lie group.

Using Yamabe’s idea, one can see that a counterexample to this conjecture must contain a copy of p-adic integers. John Pardon proved p-adic integers cannot act faithfully on a 3-manifold which led to a proof of Hilbert-Smith conjecture for 3-manifolds. I will talk about the main ideas of his proof.

April 21st: Zachary Gardner

Differential forms and q-deformation

Differential forms are an essential tool across much of geometry. In this talk, we will discuss the idea of differential forms and compare perspectives coming from manifold theory and algebraic geometry. Then we will introduce the notion of q-deformation and explore how it applies to differential forms. No need for any knowledge of algebraic geometry, just some familiarity with the notion of a manifold.

May 5th: Matthew Zevenbergen

The Whitehead Manifold

I'll talk about the origins and construction of the Whitehead Manifold, along with some of its properties. In particular, I'll discuss David Gabai's proof that the Whitehead Manifold is the union of two Euclidean spaces, whose intersection is another copy of Euclidean space.

May12th: Marius Huber

Ribbon concordance and cobordism as partial orders

In this talk, I will discuss Agol's recent proof of the fact that ribbon concordance gives a partial order on the set of knots in S^{3}. Moreover, I will show how the methods he used can be used to make progress towards conjecture asserting that ribbon cobordism gives a partial order on the set of 3-manifolds.